Solutions-Chapter11

Post by **answerhappygod** » Tue Aug 03, 2021 7:45 am

Chapter 11
Optimal Portfolio Choice and the Capital Asset Pricing Model
11-2. You own three stocks: 600 shares of Apple Computer, 10,000 shares of Cisco Systems, and 5000 shares of Colgate-Palmolive. The current share prices and expected returns of Apple, Cisco, and Colgate-Palmolive are, respectively, $500, $20, $100 and 12%, 10%, 8%.
9
IN
boo x
Soo
=
300,000
AMOUNT AMOUNT
IN APPLE
=
a. What are the portfolio weights of the three stocks in your portfolio?
b. What is the expected return of your portfolio?
IN
Cisco Colgate
10000×20
200
000
AM WNT ==
=
=
,
=
Sooo x too =
500,0001
ooo ,,
ooo
=
TOTAL
X
1 E ( R p) -
3,06%-0=030 Xc ,
200002 1000000
.
20
→
,
Apple
X
, soo
50000L
so
Colgate =. 1000000
=
-
11-4.
There are two ways to calculate the expected return of a portfolio: either calculate the expected return using the value and dividend stream of the portfolio as a whole, or calculate the weighted average of the expected returns of the individual stocks that make up the portfolio. Which return is higher?
Both calculations of expected return of a portfolio give the same answer.
X .
Xaswt (to)-1.50to8)=
ECR 30612) t
t iho
Apple =
Apple)
Rasco t Xcoya,zE(RAYATE ()
.096=9 . 6%
)

11-5. Using the data in the following table, estimate (a) the average return and volatility for each stock, (b) the covariance between the stocks, and (c) the correlation between these two stocks.
􏰀10􏰁20􏰁5􏰀5􏰁2􏰁9 a. RA 􏰂 6
Variance of A 􏰂
􏰂 0.01123
Volatility of A = SD(RA ) 􏰂 Variance of A 􏰂 .01123 􏰂 10.60%
RB 􏰂
􏰂 12%
Variance of B 􏰂
􏰂 0.02448
􏰂3.5%
1􏰇􏰃0.2􏰀0.035􏰄2 􏰁􏰃0.05􏰀0.035􏰄2 􏰁 􏰈 􏰇 􏰈
21􏰁30􏰁7􏰀3􏰀8􏰁25 6
􏰅􏰃􏰀0.1􏰀0.035􏰄2 􏰁 􏰆 􏰇􏰈
5􏰇􏰃􏰀0.05􏰀0.035􏰄2 􏰁􏰃0.02􏰀0.035􏰄2􏰈 􏰇􏰈
􏰇􏰁 􏰃0.09 􏰀 0.035􏰄2 􏰈 􏰉􏰊
􏰅􏰃0.21􏰀0.12􏰄2 􏰁􏰃0.3􏰀0.12􏰄2 􏰁 􏰆
1􏰇22􏰈 􏰇􏰃0.07 􏰀 0.12􏰄 􏰁 􏰃􏰀0.03 􏰀 0.12􏰄 􏰁􏰈 5􏰇􏰈
􏰇􏰃􏰀0.08􏰀0.12􏰄2 􏰁􏰃0.25􏰀0.12􏰄2 􏰈 􏰉􏰊
Volatility of B = SD(RB ) 􏰂 Variance of B 􏰂 􏰅􏰃􏰀0.1􏰀 0.035􏰄􏰃0.21􏰀 0.12􏰄􏰁
.02448 􏰂 15.65%
􏰆
􏰈 􏰇􏰈
􏰇􏰃0.2 􏰀 0.035􏰄􏰃0.07 􏰀 0.12􏰄􏰁
1􏰇􏰃0.05􏰀0.035􏰄􏰃0.30􏰀0.12􏰄􏰁 􏰈 b. Covariance􏰂 􏰇 􏰈 5􏰇􏰃􏰀0.05􏰀0.035􏰄􏰃􏰀0.03􏰀0.12􏰄􏰁􏰈
c. Correlation
􏰂 0.00104 Covariance
􏰂 􏰂 SD(RA ) SD(RB )
0.00104 (0.1060) (0.1565)
= 0.0627
􏰇 0.02􏰀0.035 􏰀0.08􏰀0.12 􏰁 􏰈 􏰇􏰃 􏰄􏰃 􏰄􏰈
􏰇􏰈
􏰇􏰃0.09 􏰀 0.035􏰄􏰃0.25 􏰀 0.12􏰄 􏰈 􏰉􏰊
11-9. Supposetwostockshaveacorrelationof1.Ifthefirststockhasanaboveaveragereturnthis year, what is the probability that the second stock will have an above average return?
Because the correlation is perfect, they move together (always), so the probability is 1.

"
"
=
"
"^"""""
§ %¥&"%k%¥¥
IN A
5 VAR CRp) XWTARIRW) t Xa ' VARGA) t 2X Xa Cor (Rw Ra
4
we ARE Investing
=
,
"^^ " "" "
11-11. Suppose Wesley Publishing’s stock has a volatility of 60%, while Addison Printing’s stock has a volatility of 30%. If the correlation between these stocks is 25%, what is the volatility of the following portfolios of Addison and Wesley: (a) 100% Addison, (b) 75% Addison and 25% Wesley, and (c) 50% Addison and 50% Wesley.
DENOTE WESLEY PUBLISHING BY ‘W’ AND ADDISON PRINTING BY ‘A’.
[email protected]) Everything
r
So ( Portfolio ) VARIRpf-C255C.to/t.C755f3oJt2(.25-)C75)C.z5)E6o)C3o)=
.
of .
SDRa =3
(
)
⇐
w
)
His 't ⇐ DCRWD
So (Rw)Sofla)
. 30=3070
Cso ==I
11-20. You currently hold a portfolio of three stocks, Delta, Gamma, and Omega. Delta has a volatility of 60%, Gamma has a volatility of 30%, and Omega has a volatility of 20%. Suppose you invest 50% of your money in Delta, and 25% each in Gamma and Omega.
a. What is the highest possible volatility of your portfolio?
b. If your portfolio has the volatility in (a), what can you conclude about the correlation between Delta and Omega?
a. Maximum volatility = weighted average = 0.5(60%) + 0.25(30%) + 0.25(20%) = 42.5%
b. Correlation = 1 (as there are no diversification benefits – refer to class notes)
11-21. Suppose Ford Motor stock has an expected return of 20% and a volatility of 40%, and Molson Coors Brewing has an expected return of 10% and a volatility of 30%. If the two stocks are uncorrelated,
a. Whatistheexpectedreturnandvolatilityofanequallyweightedportfolioofthetwostocks?
b. Given your answer to (a), is investing all of your money in Molson Coors stock an efficient portfolio of these two stocks?
. 09
SD (Rp)
VARI 6
= ARD=
Fog =
-5Cf . ) ) C. Cbo ) 5630RftBpfo
2 so 25 605-1 so so C )
)
FBE Co =
0135
SD (Rp)
ARD
.
3674=36.7470
.
DENOTE FORD MOTOR BY ‘F’ AND MOLSON COORS BREWING BY ‘M’.
't f
= 9E)to) )
15=1570
from QUESTION
LRp So .
Go
50
=
.
USE THE VARI
FORMULA
FOR
C Rp)
.
VAR It : 56C634¢40s00fo5t -1
so . 5C
so o (
)
=
.
0625
=== SD(Rp) Yp) 70625
.
25=2570
So R2Lpf)C. )
C so ))
.
.
I

b. No, as it is dominated by the 50-50 portfolio in Ford and Molson. The 50-50 portfolio provides a higher return (15% vs. 10%) and a lower volatility/risk (25% vs. 30%).
For Problems 23-26, suppose Johnson & Johnson and the Walgreen Company have expected returns and volatilities shown below, with a correlation of 22%.
11-23. Calculate (a) the expected return and (b) the volatility (standard deviation) of a portfolio that is equally invested in Johnson & Johnson’s and Walgreen’s stock.
In this case, the portfolio weights are xj = xw = 0.50. From Eq. 11.3, E[ RP ] 􏰂 x j E[ R j ] 􏰁 xw E[ Rw ]
􏰂 0.50(7%) 􏰁 0.50(10%) 􏰂 8.5%.
We can use Eq. 11.9.
SD(R )􏰂 x 2SD(R )2 􏰁x 2SD(R )2 􏰁2x x Corr(R ,R )SD(R )SD(R ) Pjjwwjwjwjw
􏰂 0.502 (0.162 ) 􏰁 0.502 (0.20)2 􏰁 2(0.50)(0.50)(0.22)(0.16)(0.20) 􏰂 14.1%
11-24. For the portfolio in Problem 23, if the correlation between Johnson & Johnson’s and Walgreen’s stock were to increase,
a. Would the expected return of the portfolio rise or fall?
b. Would the volatility of the portfolio rise or fall?
a. The expected return would remain constant, assuming only the correlation changes, 0.5 􏰋 0.07 + 0.5 􏰋 0.10 = 0.085.
b. The volatility of the portfolio would increase (due to the correlation term in the equation for the volatility of a portfolio).
11-25. Calculate (a) the expected return and (b) the volatility (standard deviation) of a portfolio that consists of a long position of $10,000 in Johnson & Johnson and a short position of $2000 in Walgreen’s.
In this case, the total investment is $10,000 – 2,000 = $8,000, so the portfolio weights are xj = 10,000/8,000 = 1.25, xw = –2,000/8,000 = –0.25. From Eq. 11.3,
E[ RP ] 􏰂 x j E[ R j ] 􏰁 xw E[ Rw ]
􏰂 1.25(7%) 􏰀 0.25(10%)
􏰂 6.25%.
We can use Eq. 11.9,
SD(R )􏰂 x 2SD(R )2 􏰁x 2SD(R )2 􏰁2x x Corr(R ,R )SD(R )SD(R ) Pjjwwjwjwjw
􏰂 1.252 (0.162 ) 􏰁 (􏰀0.25)2 (0.20)2 􏰁 2(1.25)(􏰀0.25)(0.22)(0.16)(0.20) 􏰂 19.5%.

11-34. You have $100,000 to invest. You choose to put $150,000 into the market by borrowing $50,000.
a. If the risk-free interest rate is 5% and the market expected return is 10%, what is the
expected return of your investment?
b. If the market volatility is 15%, what is the volatility of your investment? ON
THIS PROBLEM IS SIMILAR To THE EXAMPLE ON Buying on MARGIN
PAYE 13 of THE CLASS NOTES for CHAPTER It . WE WILL USE THE FORMULAE ''
from
9 I Soooo
'
.
-
PAGE
12 of
CLASS Soooo
NOTES
IN Vf -
DENOTE
TOTAL -
MARKET Investment
IN
,
; Egg
I Soooo =
t
m=
5SD (Rxm)
t 5)
solo
. 125=12570 -
-
M
-
-
YE )
=
5 =
1.5
Worf
.
,
=
SDLRm
1.5 1570
.
=
Way )()
22570
11-37. Assume all investors want to hold a portfolio that, for a given level of volatility, has the maximum possible expected return. Explain why, when a risk-free asset exists, all investors will choose to hold the same portfolio of risky stocks.
Investors who want to maximize their expected return for a given level of volatility will pick portfolios that maximize their Sharpe ratio. The set of portfolios that do this is a combination of a risk-free asset and a single portfolio of risk assets—the tangential portfolio.
11-48. Suppose the risk-free return is 4% and the market portfolio has an expected return of 10% and a volatility of 16%. Merck & Co. (Ticker: MRK) stock has a 20% volatility and a correlation with the market of 0.06.
a. What is Merck’s beta with respect to the market?
b. Under the CAPM assumptions, what is its expected return?
CLASS NOTES
of 4THE ) )
At THE Bottom
B SD CRM trek X CORR CRM true Rmkt
.
USE FORMULA =
,
(0-207606)--0.075
By
M.
PAGE 17 of
.
( -
)
50000 too ooo
Wm
E(Rx)to540% C-
=
-
=
mere , ,
1 E(Rm Era ) =
# SD IRmkt)
r 04+0.025 flo -
=
. 16 @)
11-50. Suppose Autodesk stock has a beta of 2.16, whereas Costco stock has a beta of 0.69. If the risk-free interest rate is 4% and the expected return of the market portfolio is 10%, what is the expected return of a portfolio that consists of 60% Autodesk stock and 40% Costco stock, according to the CAPM?
􏰌p 􏰂􏰃0.6􏰄􏰃2.16􏰄􏰁􏰃0.4􏰄􏰃0.69􏰄􏰂1.572
E􏰅R 􏰆􏰂4􏰁􏰃1.572􏰄􏰃10􏰀4􏰄 􏰂13.432% 􏰉p􏰊
.
043=0.0445--4.4570